A class of parameter choice strategies which, include an approximated version of the Arcangeli's method is considered for ill-posed operator equations when the data are not known exactly. A particular case of the method is a generalization and modification of the Marti's method, in which case the results include, and in certain cases improve the conclusions of Engl and Neubauer (1985) under weaker conditions.

Thamban Nair M

Department of Mathematics

Approximation theory

Mathematical operators

ARCANGELI'S METHOD

MARTI'S METHOD

TIKHONOV REGULARIZATION

Numerical methods

IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

On a generalized arcangeli's method for tikhonov regularization with inexact data

Numerical Functional Analysis and Optimization

While applying regularization procedures for obtaining stable solutions of ill-posed problems, one of the crucial step is the choice of the regularization parameter. Among the well considered discrepancy principles in the literature, Morozov's method and Arcangeli's method are widely used because of their simplicity for the purpose of applications. Although Morozov's method and their variations have been considered extensively in the literature for general class of regularization methods, the Arcangeli's method is known to have applied only for Tikhonov regularization. The reason could be the belief that it can never yield a rate better than Morozov's procedure, under any smoothness assumption on the solution. However, this belief was misplaced as it has been showed by Nair (1992) that Arcangeli's method do provide the best rate O(δ2/3) for Tikhonov regularization under sufficient smoothness assumption on the solution, while Morozov's method gives the rate only up to O(δ1/2). The purpose of this paper is to consider a generalized form of Arcangeli's method for a general class of regularization methods for the case when there is no error on the modeling, and then extend the procedure which allow error in the modeling as well.

Fulltext

Journal of Inverse and Ill-Posed Problems

Generalized Arcangeli's discrepancy principles for a class of regularization methods for solving ill-posed problems

Pereverzev (1995) considered Tikhonov regularization combined with a modified form of a projection method for obtaining stable approximate solutions for ill-posed operator equations. He showed, under a certain a priori choice of the regularization parameter and a specific smoothness assumption on the solution, that the method yields the optimal order with less computational information, in the sense of complexity, than the projection method considered by Plato and Vainikko (1990). In this paper we apply a modified form of the Arcangeli's discrepancy principle for choosing the regularization parameter, and show that the conclusions of Pereverzev still hold. In fact, we do the analysis using a modified form of the generalized Arcangeli's method suggested by Schock (1984) under more flexible smoothness assumption on the solution, as has been done by George and Nair (1998), and derive the optimal result as a special case. Moreover, we compare the computational complexity of the present method with two tradit ional projection methods in the case of a priori parameter choice, and also discuss the computational complexity required to implement the suggested discrepancy principle.

Arcangeli's discrepancy principle for a modified projection scheme for ill-posed problems

On a generalized Arcangeli's method for Tikhonov regularization with Inexact data

Journal	Numerical Functional Analysis and Optimization
Publisher	Marcel Dekker Inc.
ISSN	01630563
Open Access	No